Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C2×#23530 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C2×#121 |
| prime radical | C22×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4682 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(3,4)(5,6)(7,8)(9,19)(10,24)(11,16)(12,17)(13,18)(14,22)(15,27)(20,29)(21,31)(23,30)(25,37)(26,33)(28,32)(34,35)(36,38);(7,9)(8,14)(10,25)(11,21)(12,22)(13,23)(15,29)(16,19)(17,30)(18,31)(20,28)(24,34)(26,36)(27,35)(32,38)(33,37);(4,6)(7,10,11,26,13,28,12,27)(8,15,17,32,18,33,16,24)(9,20,22,36,23,25,21,35)(14,29,19,34,31,37,30,38);(7,11,13,12)(8,16,18,17)(9,21,23,22)(10,26,28,27)(14,19,31,30)(15,24,33,32)(20,35,25,36)(29,34,37,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339136);